Problem Solving

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Brief Explanation

(Alberta Program of Studies)

Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type How would you...? or How could you...?, the problem-solving approach is being modelled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies.
A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement.
Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. Creating an environment where students openly look for, and engage in, finding a variety of strategies for solving problems empowers students to explore alternatives and develops confident, cognitive mathematical risk takers.

Research



Why Teach through Problem Solving?
  • Math makes more sense: When the problem is a “real-life” one, students have the chance to build essential connections between what the math is, why it is needed, and how it is applied.


  • A problem solving approach provides the teacher with better insight into students’ mathematical thinking. A problem solving approach provides the teacher with useful information to improve his or her mathematical interactions with students. The teacher is able to see how effectively students can reach into their “mathematical tool box” to choose the right tools, as well as see how effectively they can use those tools.


  • Problems are more motivating when they are appropriately challenging. Although some students are comfortable with being told how to do something and then doing it over and over, many do not enjoy this approach. Most students prefer a manageable challenge.


  • Problem solving builds perseverance. Many students think that if they cannot answer a math question instantly, it is too hard. Through problem solving experience, they build up a willingness to preserve at solving a problem.


  • Problem solving builds confidence, maximizes the potential for understanding and allows for differences in style and approach. Problem solving allows each individual the opportunity to create his or her own path through mathematics.


  • Problems can provide practice, both with concepts and with skills. Many good problems have the potential to ensure that students learn concepts and also have the opportunity to practice valuable skills.


  • A problem solving approach provides students with better insight into what mathematics is all about. Math requires the same kind of struggle as does creating a new piece of writing or a new work of art. Most rich problems invite many possible solution strategies, and some are even designed to encounter many possible answers. This notion of creativity or choice in mathematics will not make sense to someone who has not struggles on his or her own to try to solve a math problem.

Students need to practice problem solving. If the goal of mathematics education is to enable students to confront new situations involving mathematics, then they must practice doing this. Students need many opportunities to practice problem solving to be able to do so independently.

Characteristics of a Numerate Individual

Clearly Identified Key Outcomes

1) Number Sense


2) Patterns and Relations

  • Patterns
  • Variables and Equations

3) Shape and Space

  • Measurement
  • 3-D Objects and 2-D Shapes
  • Transformations

4) Statistics and Probability

  • Data Analysis
  • Chance and Uncertainty

Balanced Assessment Practices


Problem Solving
Criteria
 Level 1
 Level 2
 Level 3
 Level 4
Selects, sequences and applies mathematical processes appropriate to the task.
 Selects, sequences and applies mathematical processes to the assigned task with significant prompting.
 Selects, sequences and applies mathematical processes to the assigned task with minimal prompting.
 Selects, sequences and applies mathematical processes to the assigned task independently.
 Selects, sequences and applies mathematical processes to the assigned task independently with a broader view of the task.
Uses critical thinking skills to solve a problem.
 Uses minimal logic and precision in mathematical reasoning to solve problems.
 Uses logic to solve problems but lacks precision in mathematical reasoning.
 Solves problems logically and with precision in mathematical reasoning.
 Demonstrates a sophisticated level of mathematical reasoning and precision in solving problems.


Purposeful Instructional Strategies

Personalization of Learning